Oh, I thought you were making some argument about the ordering of the doors, which is of course arbitrary. But people make weird mistakes sometimes.

Yeah, my bad...

I meant 3rd as in the 1st being the door you chose, the 2nd being the door you didn't choose but had the prize (under the assumption that you chose a door without a prize) and therefore the 3rd being the unchosen, non-prize carrying door, which would have to be opened by necessity.

Similarly, take hazel-rah's coin formulation of this problem. You get a 1/3 answer if you assume that we only play the game if hazel-rah first ascertains that at least one coin is heads (this is encoded in hazel-rah's program by first eliminating all families with two daughters) and will always reveal a heads-up coin.

If instead you assume that the game will always be played, and the coin is being chosen randomly, then there is a chance that hazel-rah will uncover a coin and say that it is tails. In that case, the probability that hazel-rah uncovers a head is 0 in the TT case, 1/2 in the HT case, 1/2 in the TH case and 1 in the HH case. So if we play the game and hazel-rah uncovers a head, the probability of the other coin being heads goes back to 1/2.

It is not necessary to assume that we only play if there is at least one heads. We know there is one heads because he showed it to us. What we need to know in order to determine whether the probability of HH is 1/2 or 1/3 is whether he is preferentially showing us heads. If he looks at the coins, and chooses a heads coin to show us, the probability of HH is 1/3. If he randomly shows us a coin, and it happens to be heads, the probability of HH is 1/2. It is also possible that he preferentially shows tails, in which case the probability of HH is 1.

Alamout wrote:

Quote:

Wait, is this even true? Why does 1 coin being heads have any bearing on the other coin?

It's not true, hazel-rah is wrong. The problem is that he or she should count the heads-heads case twice, as was mentioned earlier (you could have been shown either one of those coins).

This is only true if he is selecting the coin to show you randomly, rather than looking and choosing. The way he phrased it, it is ambiguous. If he is looking and choosing a heads coin to show you, then his analysis is correct.

This is only true if he is selecting the coin to show you randomly, rather than looking and choosing. The way he phrased it, it is ambiguous. If he is looking and choosing a heads coin to show you, then his analysis is correct.

I meant that his explanation was wrong as an answer to the question in the OP.

My initial angle was that the way the OP had phrased the problem made it non-ambiguous compared to the way that the Wikipedia article phrased it. But I'm thinking Vishnu is right now, and the ambiguity is still there. His coded example made it clear to me!

Birthrates are not 50/50 between male and female. 107 boys are born worldwide for every 100 girls that are born. (Even if you completely factor out selective abortions, the ratio is still not 50/50)

Yes, there are 4 possibilities: G-G, G-B, B-G, and B-B, however you have to throw away the possibilities that aren't possible since person 1 is male, leaving only 2 possibilities. It is disingenuous to allow the boy to be either person 1 or person 2 within the same analysis of possibilities. They must be exclusively considered as either person 1 or person 2 throughout any particular atomic analysis to be valid.

However, your coworker is trying to be friendly with you, and you should respond with appropriate social graces:

If you are just getting to know each other, go find other similar brain teasers and share them. Bonus points if you tell them the wrong answer is the right one and you get them to believe those as well.

If you already have a solid friendship, take them out drinking. After you accumulate a large bar tab, offer to "bet" on either that puzzle or on a similar puzzle. If he is right, you pay. If he is wrong, he pays. Make sure you pick a drinking establishment where you can justify/verify that he is wrong via a website you can reach on your cell-phone or mobile device. Bonus points if you pick a puzzle that has quasi-logical explanations of why multiple different and conflicting answers are the "right" answers at various web-links so that no matter what they say you can choose to either let them win, or make them pay your bar tab simply by which "answer" link you pick.

There does appear to be a biological bias in sex ratio after factoring out human intervention (see: Human sex ratio). Exactly what's going on is still being studied, but you can imagine that over an evolutionary time scale that it might be advantageous to make a few more male babies because they die more. The sex ratio of babies is less important than the ratio at sexual maturity, and so you might bias the initial ratio to get 50/50 at some later age.

That said, bringing the 107/100 ratio into this question is one of those "I read Wikipedia and feel like being obnoxious" pedantic nerd things to do. It's a word problem, not an actual question. Assuming that the implied ratio is 50/50 is the normal, socialized-person thing to do.

The 107/100 ratio isn't even relevant to the question, as presumably the asker's sibling was not born yesterday. The sex ratio is biased the other way after a while, and the overall ratio (in the US) is 97 males to 100 females. The "most correct"* way to actually estimate the probability would be to establish some probability distribution for the age of the unknown sibling (perhaps based on the age of the asker) and then estimate the sex ratio based on a) the probability that the sibling is of a given age and b) the sex ratio at that age.

* Again, this would be an obnoxiously nerdy approach to what was meant to be a brain teaser.

The "most correct"* way to actually estimate the probability would be to establish some probability distribution for the age of the unknown sibling (perhaps based on the age of the asker) and then estimate the sex ratio based on a) the probability that the sibling is of a given age and b) the sex ratio at that age.

You could consider a distribution over ages and then the probability of male/female conditioned on each age, but you don't actually need to do that. The easiest empirical solution is to consider each family of exactly two siblings and for each, choose one uniformly at random to be yourself. If you're male, categorize the other sibling by gender; if you're female, discard this family from consideration. Then count the resulting number of male vs. female siblings. Such a method should take into account age, mortality, and whatever other factors there are.

If you have such information, sure. It sounds fairly difficult to acquire--you need the actual makeup of all US families. As you use more information to filter (i.e. if you know the age of the asker) then eventually you'll just find the asker's family and check what gender their sibling is. But that's no longer about probability, it's just data-mining.

Not that I have data on male/female ratios across ages or data about sibling ages, but it seems more plausible to me that such data is available. At the very least you could approximate such data with known data about ratios at various ages (birth, teenagers, senior citizens, whatever).

Don't forget to take into account the probability that a family has exactly two siblings conditioned on the age of one sibling. That is, the distribution of ages of two siblings in a family is different than the distribution of ages of people. So things could get too complex to do with simple publicly available stats.

That said, bringing the 107/100 ratio into this question is one of those "I read Wikipedia and feel like being obnoxious" pedantic nerd things to do. It's a word problem, not an actual question. Assuming that the implied ratio is 50/50 is the normal, socialized-person thing to do.

My initial angle was that the way the OP had phrased the problem made it non-ambiguous compared to the way that the Wikipedia article phrased it. But I'm thinking Vishnu is right now, and the ambiguity is still there. His coded example made it clear to me!

The way the OP phrased the problem does make it non-ambiguous. The fact that the OP's co-worker is forced to be himself, and cannot elect to be his brother, removes any ambiguity. In particular it removes any potential for dependence of the unknown sibling's gender on the known's - leaving the simple 50/50 chance of a single child being born male, without regard for the gender of the known sibling.

Don't forget to take into account the probability that a family has exactly two siblings conditioned on the age of one sibling. That is, the distribution of ages of two siblings in a family is different than the distribution of ages of people. So things could get too complex to do with simple publicly available stats.

I skimmed through the posts, but to be even more explicit, without specifying order in the cases (i.e. the designated male is the first OR the second) then the probability cases collapse to:

Two GirlsOne Girl, One BoyTwo Boys

specifying a Boy eliminates the first case. The probability of 1/3 counts the "one of each" case twice. If order is specified, then either the boy-first mixed case or boy-second mixed case is eliminated, still leaving 1/2.

Don't forget to take into account the probability that a family has exactly two siblings conditioned on the age of one sibling. That is, the distribution of ages of two siblings in a family is different than the distribution of ages of people. So things could get too complex to do with simple publicly available stats.

I skimmed through the posts, but to be even more explicit, without specifying order in the cases (i.e. the designated male is the first OR the second) then the probability cases collapse to:

Two GirlsOne Girl, One BoyTwo Boys

specifying a Boy eliminates the first case. The probability of 1/3 counts the "one of each" case twice. If order is specified, then either the boy-first mixed case or boy-second mixed case is eliminated, still leaving 1/2.

But in the case of the OP's question order is specified by the fact that his co-worker cannot choose which child he is, he must be himself.